Multiplicative order

Builds on Congruence modulo n Coprime integers Euler's theorem

Definition

Let $n$ be a positive integer and let $a$ be an integer coprime to $n$ . The multiplicative order of $a$ modulo $n$ is the smallest positive integer $k$ such that $a^k \equiv 1 \pmod n$ . The notation for it is $\operatorname{ord}_n(a)$ .

Remarks

The order exists, because $a^{\varphi(n)} \equiv 1 \pmod n$ by Euler's theorem. For each integer $m \ge 0$ , $a^m \equiv 1 \pmod n$ if and only if $\operatorname{ord}_n(a) \mid m$ . In particular, $\operatorname{ord}_n(a) \mid \varphi(n)$ . The order of $a$ modulo $n$ is the order of $\overline{a}$ in the group $(\mathbb{Z}/n\mathbb{Z})^\times$ . For example, $\operatorname{ord}_7(2) = 3$ , because $2^1 = 2$ , $2^2 = 4$ and $2^3 = 8 \equiv 1 \pmod 7$ .

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Divisibility definition
Congruence modulo n definition
Greatest common divisor definition
Coprime integers definition
Euler's totient function definition
Euler's theorem statement
Multiplicative order definition

Where this leads

6 pages build on this one.

Tag	`0SBRX5CZ5R`
URI	`m2://0SBRX5CZ5R`
Aliases	`order-modulo-n`
Kind	definition
Writing	W1 stated
MSC	11A07
Defines	Multiplicative order
Uses	Congruence modulo n, Coprime integers, Euler's theorem

Multiplicative order

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

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